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<title>Spheroidal wave function</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spheroidal wave function</span></span>
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<p><b>Spheroidal wave functions</b> are solutions of the <a href="Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a> that are found by writing the equation in spheroidal coordinates and applying the technique of <a href="Separation_of_variables" title="Separation of variables">separation of variables</a>, just like the use of <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> lead to <a href="Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. They are called <i>oblate spheroidal wave functions</i> if <a href="Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">oblate spheroidal coordinates</a> are used and <i><a href="Prolate_spheroidal_wave_functions" class="mw-redirect" title="Prolate spheroidal wave functions">prolate spheroidal wave functions</a></i> if <a href="Prolate_spheroidal_coordinates" title="Prolate spheroidal coordinates">prolate spheroidal coordinates</a> are used.<sup id="cite_ref-Flammer1957_1-0" class="reference"><a href="#cite_note-Flammer1957-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
If instead of the Helmholtz equation, the <a href="Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions
are called <i>oblate harmonics</i> and with prolate spheroidal coordinates, <i>prolate harmonics</i>. Both type of spheroidal harmonics
are expressible in terms of <a href="Legendre_functions" class="mw-redirect" title="Legendre functions">Legendre functions</a>.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Oblate_spheroidal_coordinates" title="Oblate spheroidal coordinates">Oblate spheroidal coordinates</a>, especially the section <a href="Oblate_spheroidal_coordinates#Oblate_spheroidal_harmonics" title="Oblate spheroidal coordinates"><i>Oblate spheroidal harmonics</i></a>, for a more extensive discussion.</li>
<li><a href="Oblate_spheroidal_wave_function" title="Oblate spheroidal wave function">Oblate spheroidal wave function</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<dl><dt>Notes</dt></dl>
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<li id="cite_note-Flammer1957-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Flammer1957_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFFlammer,_C.1957" class="citation book cs1">Flammer, C. (1957). <i>Spheroidal wave functions</i>. Stanford University Press Stanford, Calif.</cite></span>
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<dl><dt>Bibliography</dt></dl>
<ul><li>C. Niven <i>On the Conduction of Heat in Ellipsoids of Revolution.</i> Philosophical transactions of the Royal Society of London, v. 171 p. 117 (1880)</li>
<li>M. Abramowitz and I. Stegun, <i>Handbook of Mathematical function</i> (US Gov. Printing Office, Washington DC, 1964)</li>
<li><cite id="CITEREFVolkmer2010" class="citation cs2">Volkmer, H. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/30">"Spheroidal wave function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li></ul>
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